Transcript Ch 33 - A.C. Circuits

Alternating Current Circuits
Chapter 33
(continued)
Phasor Diagrams
A phasor is an arrow whose length represents the amplitude of
an AC voltage or current.
The phasor rotates counterclockwise about the origin with the
angular frequency of the AC quantity.
Phasor diagrams are useful in solving complex AC circuits.
Resistor
Capacitor
Inductor
Vp
Ip
Vp
Ip
wt
Ip
wt
wt
Vp
Reactance - Phasor Diagrams
Resistor
Capacitor
Inductor
Vp
Ip
Vp
Ip
wt
Ip
wt
wt
Vp
“Impedance” of an AC Circuit
R
L
~
C
The impedance, Z, of a circuit relates peak
current to peak voltage:
Ip 
Vp
Z
(Units: OHMS)
“Impedance” of an AC Circuit
R
L
~
C
The impedance, Z, of a circuit relates peak
current to peak voltage:
Ip 
Vp
Z
(Units: OHMS)
(This is the AC equivalent of Ohm’s law.)
Impedance of an RLC Circuit
R
E
~
L
C
As in DC circuits, we can use the loop method:
E - V R - VC - VL = 0
I is same through all components.
Impedance of an RLC Circuit
R
E
~
L
C
As in DC circuits, we can use the loop method:
E - V R - VC - VL = 0
I is same through all components.
BUT: Voltages have different PHASES
 they add as PHASORS.
Phasors for a Series RLC Circuit
Ip
VLp
VRp
f
(VCp- VLp)
VP
VCp
Phasors for a Series RLC Circuit
Ip
VLp
VRp
f
(VCp- VLp)
VP
VCp
By Pythagoras’ theorem:
(VP )2 = [ (VRp )2 + (VCp - VLp)2 ]
Phasors for a Series RLC Circuit
Ip
VLp
VRp
f
(VCp- VLp)
VP
VCp
By Pythagoras’ theorem:
(VP )2 = [ (VRp )2 + (VCp - VLp)2 ]
= Ip2 R2 + (Ip XC - Ip XL) 2
Impedance of an RLC Circuit
R
Solve for the current:
Ip 
Vp
Vp

Z
R2  (X c  X L )2
~
L
C
Impedance of an RLC Circuit
R
Solve for the current:
Ip 
~
L
C
Vp

Z
R2  (X c  X L )2
Impedance:
Vp
Z
 1

R 
 wL
wC
2
2
Impedance of an RLC Circuit
Vp
Ip 
Z
 1
2
R 
 wL
wC
Z
The current’s magnitude depends on
the driving frequency. When Z is a
minimum, the current is a maximum.
This happens at a resonance frequency:
2
The circuit hits resonance when 1/wC-wL=0: w r=1/ LC
When this happens the capacitor and inductor cancel each other
and the circuit behaves purely resistively: IP=VP/R.
IP
R =10W
L=1mH
C=10mF
R = 1 0 0 W
0
1 0
wr
2
1 0
3
1 0
4
1 0
5
w
The current dies away
at both low and high
frequencies.
Phase in an RLC Circuit
Ip
VLp
We can also find the phase:
VRp
(VCp- VLp)
f
VP
tan f = (VCp - VLp)/ VRp
or;
or
VCp
tan f = (XC-XL)/R.
tan f = (1/wC - wL) / R
Phase in an RLC Circuit
Ip
VLp
We can also find the phase:
VRp
(VCp- VLp)
f
VP
tan f = (VCp - VLp)/ VRp
or;
or
VCp
tan f = (XC-XL)/R.
tan f = (1/wC - wL) / R
More generally, in terms of impedance:
cos f  R/Z
At resonance the phase goes to zero (when the circuit becomes
purely resistive, the current and voltage are in phase).
Power in an AC Circuit
V
f= 0
p
I
2p
wt
V(t) = VP sin (wt)
I(t) = IP sin (wt)
(This is for a purely
resistive circuit.)
P
P(t) = IV = IP VP sin 2(wt)
Note this oscillates
twice as fast.
p
2p
wt
Power in an AC Circuit
The power is P=IV. Since both I and V vary in time, so
does the power: P is a function of time.
Use, V = VP sin (wt) and I = IP sin (w t+f ) :
P(t) = IpVpsin(wt) sin (w t+f )
This wiggles in time, usually very fast. What we usually
care about is the time average of this:
1 T
P   P( t )dt
T 0
(T=1/f )
Power in an AC Circuit
Now: sin(wt  f )  sin(wt )cos f  cos(wt )sin f
Power in an AC Circuit
Now: sin(wt  f )  sin(wt )cos f  cos(wt )sin f
P( t )  I PVP sin( w t )sin( w t  f )
 I PVP sin 2( w t )cos f  sin( w t )cos( w t )sin f
Power in an AC Circuit
Now: sin(wt  f )  sin(wt )cos f  cos(wt )sin f
P( t )  I PVP sin( w t )sin( w t  f )
 I PVP sin 2( w t )cos f  sin( w t )cos( w t )sin f
Use:
and:
So
sin (w t ) 
2
1
2
sin(w t ) cos(w t )  0
P 
1
2
I PV P cos f
Power in an AC Circuit
Now: sin(wt  f )  sin(wt )cos f  cos(wt )sin f
P( t )  I PVP sin( w t )sin( w t  f )
 I PVP sin 2( w t )cos f  sin( w t )cos( w t )sin f
Use:
and:
So
sin (w t ) 
2
1
2
sin(w t ) cos(w t )  0
P 
1
2
I PV P cos f
which we usually write as
P  IrmsVrms cos f
Power in an AC Circuit
P  IrmsVrms cos f
(f goes from -900 to 900, so the average power is positive)
cos(f) is called the power factor.
For a purely resistive circuit the power factor is 1.
When R=0, cos(f)=0 (energy is traded but not dissipated).
Usually the power factor depends on frequency.
Power in an AC Circuit
P  IrmsVrms cos f
What if f is not zero?
I
P
V
Here I and V are 900
out of phase. (f 900)
wt (It is purely reactive)
The time average of
P is zero.
Transformers
Transformers use mutual inductance to change voltages:
N2
V2 
V1
N1
N1 turns
Iron Core
V1
Primary
Power is conserved, though:
(if 100% efficient.)
N2 turns
V2
Secondary
I1V1  I 2V2
Transformers & Power Transmission
Transformers can be used to “step up” and “step
down” voltages for power transmission.
110 turns
Power
=I1 V1
V1=110V
20,000 turns
V2=20kV Power
=I2 V2
We use high voltage (e.g. 365 kV) to transmit electrical
power over long distances.
Why do we want to do this?
Transformers & Power Transmission
Transformers can be used to “step up” and “step down”
voltages, for power transmission and other applications.
110 turns
Power
=I1 V1
V1=110V
20,000 turns
V2=20kV Power
=I2 V2
We use high voltage (e.g. 365 kV) to transmit electrical
power over long distances.
Why do we want to do this?
P = I2R
(P = power dissipation in the line - I is smaller at high voltages)